Maps of linear systems on blow-ups of the projective plane

Journal of Pure and Applied Algebra 156 (2001) 1–14

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Maps of linear systems on blow-ups of the projective plane Stephanie Fitchett ∗ Department of Mathematics, Duke University, Durham, NC 27708-0320, USA Received 4 September 1998; received in revised form 19 May 1999 Communicated by A.V. Geramita

Abstract Let X be the blow-up of P2 at n points p1 ; : : : ; pn in linearly general position, let F be a numerically eective divisor on X , and let L be the total transform on X of a line on P2 . The natural multiplication map (X; OX (F)) ⊗ (X; OX (L)) → (X; OX (F + L)) is shown to be surjective for a particular subset of numerically eective divisors. This result implicitly determines minimal free resolutions of ideals deÿning fat point subschemes supported at six or fewer general c 2001 Elsevier Science B.V. points of P2 . All work is over an algebraically closed ÿeld k. All rights reserved. MSC: 14; 13

1. Introduction Let X ⊂ Pr be a projective variety. For any two coherent sheaves F and G on X , there is a natural multiplication map (X; F) ⊗ (X; G) → (X; F ⊗ G);

(1)

where is the global section functor. Such maps on tensor products of global are useful in many contexts, including the study of resolutions of ideals deÿning the variety X (see [2,4], for instance), and, more generally, for computing certain Koszul cohomology groups [5]. Our interest in (1) is when X is the blow-up of P2 at n points p1 ; : : : ; pn ; F=OX (F) and G = OX (L); where F is an eective divisor on X and L is the pullback to X of a ∗ Current address: Honors College, Florida Atlantic University, Jupiter, FL 33458-8888, USA. E-mail address: sÿ[email protected] (S. Fitchett).

c 2001 Elsevier Science B.V. All rights reserved. 0022-4049/01/$ - see front matter PII: S 0 0 2 2 - 4 0 4 9 ( 9 9 ) 0 0 1 1 5 - 2

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line on P2 . In this situation the ranks of the maps F : (X; OX (F)) ⊗ (X; OX (L)) → (X; OX (F + L)); for various F, allow us to determine the number of elements of each degree in minimal generating sets for ideals deÿning fat point subschemes Z = m1 p1 + · · · + mn pn of P2 . Because minimal free resolutions for height 2 perfect ideals I have the form 0 → F1 → F0 → I → 0; determining the number of elements of each degree in a minimal generating set for I implicitly determines the modules in a minimal free resolution of I (the generators determine F0 and the Hilbert functions of F0 and I determine the Hilbert function of F1 , which in turn determines F1 ). If Z ⊂ P2 is a fat point subscheme, the degree t component of its deÿning ideal I (Z) is naturally identiÿed with a line bundle OX (Ft ) on the surface X obtained by blowing up P2 at the points p1 ; : : : ; pn . The number of elements of degree t in a minimal set of homogeneous generators for I (Z) is given by the dimension of the cokernel of the map Ft−1 (since it turns out that Ft−1 + L = Ft ). In [12], Harbourne showed that to determine the dimension of the cokernel of F for any eective divisor F, it is sucient to determine the dimension of the cokernel of H for every numerically eective divisor H . He then showed that for fat point subschemes supported at any ÿve or fewer points of P2 ; H is surjective whenever H is numerically eective. His results implicitly determine minimal free resolutions for ideals deÿning fat point subschemes of P2 supported at any 5 or fewer points. Catalisano’s results in [1] had previously allowed determination of resolutions for fat point subschemes supported at points on a smooth conic (which includes the case of 5 general points). In Section 2 we show (Theorem 2.6) that if X is the blow-up of P2 at points p1 ; : : : ; pn in linearly general position, and  is the cone of nonnegative sums of numerically eective divisor classes on X , each summand of which is naturally identiÿed with a numerically eective divisor class on a blow-up of P2 at ÿve or fewer of the points p1 ; : : : ; pn , then F is surjective when F is a numerically eective divisor in a class contained in . In Section 3, as an application of Theorem 2.6, we extend Harbourne’s and Catalisano’s results by showing that when X is the blow-up of P2 at six general points (all six points do not lie on a conic and no three of the points lie on a line) and F is a numerically eective divisor on X , although the map F may fail to be surjective, it does always have maximal rank (Theorem 3.4). For seven or more points, [12,3] give many examples in which maximal rank fails; [11] completely determines these failures for seven points; unpublished work of Harbourne, Holay and the author completely determines failures of eight general points. Such a determination remains unknown for more than eight points. The ÿnal section illustrates how Theorem 3.4 and previous results of Harbourne can be combined to ÿnd the form of a minimal free resolution for a speciÿc fat point ideal supported on six general points of P2 .

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2. Points in linearly general position To set notation, capital ital type (e.g., A) will be used to denote divisors, [A] will denote the divisor class of A, and calligraphic type (e.g., A) will be used to denote sheaves. If A is a sheaf on the k-scheme X; H i (X; A) will denote the ith sheaf cohomology and hi (X; A) will denote the k-dimension of H i (X; A). We begin with some background which will be needed for working with divisors on surfaces. If X is a closed subscheme of projective space, then for any two coherent sheaves F and G on X , there is a natural map 

(X; F) ⊗ (X; G) → (X; F ⊗ G); given by multiplication on simple tensors. Following [15], we will denote the kernel and cokernel of this map by R(F; G) and S(F; G), respectively. Proposition 2.1. Let X be a closed subscheme of projective space; let F and G be coherent sheaves on X; and let C be the sheaf associated to an eective divisor C on X . (a) If the restriction homomorphisms (X; F) → (C; F ⊗ OC ) and (X; F ⊗ G) → (C; F ⊗ G ⊗ OC ) are surjective; then we have an exact sequence 0 → R(F ⊗ C−1 ; G) → R(F; G) → R(F ⊗ OC ; G) → S(F ⊗ C−1 ; G) → S(F; G) → S(F ⊗ OC ; G) → 0: (b) If (X; G) → (C; G ⊗ OC ) is surjective; then S(F ⊗ OC ; G) = S(F ⊗ OC ; G ⊗ OC ). (c) If X is a smooth curve of genus g and F and G are line bundles of degrees at least 2g + 1 and 2g; respectively; then S(F; G) = 0. Proof. Parts 1 and 3 can be found in [15] (The 6-lemma and Theorem 6, respectively). Part 2 is stated, but not proved, in [12]. For completeness we outline a proof here. Note that the natural homomorphism (C; F ⊗ OC ) ⊗ (X; G) → (C; F ⊗ G ⊗ OC ) factors through (C; F ⊗ OC ) ⊗ (C; G ⊗ OC ). Thus we have an exact sequence 0 → A → S(F ⊗ OC ; G) → S(F ⊗ OC ; G ⊗ O|C ) → 0; and A is the cokernel of the map (C; F ⊗ OC ) ⊗ (X; G) → (C; F ⊗ OC ) ⊗ (C; G ⊗ OC ): Since A vanishes if

(X; G) → (C; G ⊗ OC ) is surjective, the result follows.

Recall that the canonical sheaf on P2 is isomorphic to OP2 (−3), and the canonical sheaf on the blow-up surface X of P2 at p1 ; : : : ; pn is OX (−3L + E1 + · · · + En ); where L is the pullback to X of a line on P2 , and Ei (1 ≤ i ≤ n) is the total transform of pi . If X → P2 is the blow-up of P2 at n distinct points p1 ; : : : ; pn , then the classes [L]; [E1 ]; : : : ; [En ] of L; E1 ; : : : ; En , respectively, form an orthogonal Z-basis of the divisor class group Cl(X ) with −1 = −[L]2 = [E1 ]2 = · · · = [En ]2 . (This basis is called an exceptional conÿguration for Cl(X ).) We will say a divisor class [F] on X is eective

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if it is the class of an eective divisor F (i.e., h0 (X; OX (F)) ¿ 0), and that [F] is numerically eective provided [F] · [E] ≥ 0 for every eective class [E]. Finally, a divisor class [C] on X is called an exceptional class if it is the class of an exceptional divisor C (i.e., C is a smooth rational curve which satisÿes C 2 = C · K = −1, where K is a canonical divisor on X ). Lemma 2.2. If F and F + G are numerically eective divisors on X with S(OX (F); OX (L)) = 0 and h1 (X; OX (F)) = 0; and G is an eective divisor with h0 (X; OX (KX + G)) = 0 and h1 (X; OX (L − G)) = 0; then S(OX (F + G); OX (L)) = 0 and h1 (X; OX (F + G)) = 0. Proof. We claim that h1 (X; OX (F))=0 and F numerically eective imply h1 (X; OX (F + L)) = 0. Consider the exact sequence H 1 (X; OX (F)) → H 1 (X; OX (F + L)) → H 1 (X; OX (F + L)|L ); since L is a rational curve and (F + L) · L ¿ 0; h1 (X; OX (F + L)|L ) = 0, forcing h1 (X; OX (F + L)) = 0. This gives the exact sequence (see Proposition 2.1) S(OX (F); OX (L)) → S(OX (F + G); OX (L)) → S(OX (F + G)|G ; OX (L)): Since h1 (X; OX (L − G)) = 0 by assumption, Proposition 2.1 shows S(OX (F + G)|G ; OX (L)|G ) ∼ = S(OX (F + G)|G ; OX (L)): Now S(OX (F + G)|G ; OX (L)|G ) = 0 by [12, Lemma 2:5], since h0 (X; OX (KX + G)) = 0. Since S(OX (F); OX (L)) = 0 by assumption, we have S(OX (F + G); OX (L)) = 0. To see that h1 (X; OX (F + G)) = 0, consider the exact sequence H 1 (X; OX (F)) → H 1 (X; OX (F + G)) → H 1 (X; OX (F + G)|G ): By [10, Corollary II.9], h1 (G; OX (F + G)|G ) = 0, and by assumption h1 (X; OX (F)) = 0. Therefore h1 (X; OX (F + G)) = 0. Notation. From now on, we will drop the OX ’s when no confusion will arise. Thus, we will write h0 (X; F) instead of h0 (X; OX (F)), etc. Also, we will nearly always consider multiplication maps F : (X; OX (F)) ⊗ (X; OX (L)) → (X; OX (F + L)), where L is the pullback to X of a line on P2 , in which case we will denote the cokernel of F by S(F) instead of the more cumbersome S(OX (F); OX (L)). Similarly, we denote the kernel of F by R(F). Let s(F) denote dimk S(F) and r(F) denote dimk R(F). We make the following deÿnition for convenience in discussing transforms of conics and lines on a blow-up X → P2 which may be neither proper nor total transforms. Deÿnition. Let C be a curve in P2 . We say the divisor D on X is a partial transform of C under a blow-up X → P2 of distinct points p1 ; : : : ; pn if D is in the class [(C · L)L − a1 E1 − · · · − an En ], where each ai is either 0 or 1, and where pi is required to lie on C if ai = 1. Note that a partial transform is just a proper transform plus (possibly) some exceptional curves.

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Corollary 2.3. Assume X is the blow-up of P2 at n distinct points. If G is a partial transform of either a conic or a line and both F and F + G are numerically eective with S(F) = 0 and h1 (X; F) = 0; then S(F + G) = 0 and h1 (X; F + G) = 0. Proof. The divisor K = −3L + E1 + · · · + En is a canonical divisor on X . Either G is in the class [2L − a1 Ei1 − · · · − an Ein ] or G is in the class [L − a1 Ei1 − · · · − an Ein ], where each ai is 0 or 1. In any case, L · (K + G) ¡ 0, so h0 (X; K + G) = 0. Using Riemann–Roch for surfaces, we can check that h1 (X; L − G) = 0. Thus Lemma 2.2 applies. Theorem 2.4. If F is a numerically eective divisor on a blow-up X of P2 at ÿve or fewer points; then S(F) = 0 and h1 (X; F) = 0. Proof. See [12] (or [1] for points in general position). We state a proposition which allows us to study Cl(X ) inductively: Proposition 2.5. Let  : X 0 → X be the blowing up of a smooth projective surface X at a point p on X. Then the induced homomorphism ∗ : Cl(X ) → Cl(X 0 ) is an inclusion which preserves eectivity and numerical eectivity of divisors. Proof. All statements are well-known. For details, a proof of the preservation of effectivity can be found in [7, Lemma 1.3]. For the preservation of numerical eectivity, see [6, Lemma 1.4]. Theorem 2.6. Assume no three of p1 ; : : : ; pn are collinear; and consider the cone  of nonnegative sums of numerically eective classes on X; each summand of which is the class of a numerically eective divisor on the blow-up of P2 at ÿve or fewer of the points p1 ; : : : ; pn . Then S(F) = 0 and h1 (X; F) = 0 for each divisor F in a class contained in . Proof. The cone  is generated by the union of the sets C0 = {[L]}; C1 = {[L − Ei1 ] | 1 ≤ i1 ≤ n}; C3 = {[2L − Ei1 − Ei2 − Ei3 ] | 1 ≤ i1 ¡ i2 ¡ i3 ≤ n}; C4 = {[2L − Ei1 − Ei2 − Ei3 − Ei4 ] | 1 ≤ i1 ¡ · · · ¡ i4 ≤ n}; and C5 = {[3L − 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5 ] | i1 ; : : : ; i5 are distinct in {1; : : : ; n}}: (See [8, Proposition I.5.3].) Let F be a numerically eective divisor on X with [F] in . If in fact [F] ∈ hC0 ; C1 i, the cone generated over the nonnegative integers by C0 ∪ C1 , then S(F) = 0 and h1 (X; F) = 0 by [10], Theorem 2:8 and Lemma 2:7, respectively. Assume that [F] ∈ hC0 ; C1 ; C3 i. For each [2L − Ei1 − Ei2 − Ei3 ] which occurs in [F], subtract one [L − Ei1 − Ei2 ]. The dierence at each stage will be numerically eective

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(since (2L − Ei1 − Ei2 − Ei3 ) − (L − Ei1 − Ei2 ) = L − Ei3 ). After doing all the subtractions, we ÿnd that any divisor in the resulting class [F 0 ] is in hC0 ; C1 i, so S(F 0 ) = 0 and h1 (X; F 0 ) = 0, as above (where F 0 is any divisor in [F 0 ]). Thus Corollary 2.3 applies and repeated use gives S(F) = 0 and h1 (X; F) = 0. Now suppose [F] ∈ hC0 ; C1 ; C3 ; C5 i. For each [3L −2Ei1 −Ei2 −Ei3 −Ei4 −Ei5 ] which occurs in [F], subtract [2L − Ei1 − Ei2 − Ei3 − Ei4 − Ei5 ]. As in the argument above, the dierence at each stage will be numerically eective. The resulting class [F 0 ] is in hC0 ; C1 ; C3 i, so S(F 0 ) = 0 by the previous case. Again repeated use of Corollary 2.3 gives S(F) = 0 and h1 (X; F) = 0. Finally, assume a class from C4 appears in [F]. We may assume [F] is not a class on a blow-up of P2 at ÿve or fewer points, since this case is covered by Theorem 2.4. Write [F] = [F1 + · · · + Fr + G] where each [Fj ] is in C4 and [G] ∈ hC0 ; C1 ; C3 ; C5 i. Without loss of generality, assume [F1 ] = [2L − E1 − E2 − E3 − E4 ]. Induct on r. The preceding argument covers the case when r = 0, so we assume r ¿ 0, and consider several cases. First, if [G] = 0, then r ≥ 2 since [F] is not a class on a blow-up of P2 at ÿve or fewer points. After reindexing we may assume [F1 ] is as before and [F2 ] = [2L − E5 − Ei2 − Ei3 − Ei4 ]. Let [F 0 ] = [F] − [2L − E1 − E2 − E3 − E4 − E5 ] = [F2 + E5 ] + [F3 + · · · + Fr ] = [2L − Ei1 − Ei2 − Ei3 ] + [F3 + · · · + Fr ]: By induction both S(F 0 ) and h1 (X; F 0 ) are 0, so applying Corollary 2.3 gives S(F) = 0 and h1 (X; F) = 0. If [G] = [L], let [F 0 ] = [F] − [L − E1 − E2 ] = [F2 + · · · + Fr ] + [2L − E3 − E4 ]. Again both S(F 0 ) and h1 (X; F 0 ) are 0 by induction, so application of Corollary 2.3 yields the required result. If [G] ∈ hC0 ; C1 ; C3 i, [G] 6= 0, [G] 6= [L], then let [F 0 ] = [F] − [2L − E1 − E2 − E3 − E4 − E5 ] = [F2 + · · · + Fr ] + [G] + [E5 ]: Since [F] is not a class on a blow-up of P2 at ÿve or fewer points, [ − E5 ] occurs in [G] or in one of the [Fi ]’s. After reindexing if need be, we may assume [ − E5 ] occurs either in [G] or in [F2 ]. In the former case, [G + E5 ] ∈ hC0 ; C1 ; C3 i, so we are done by induction and use of Corollary 2.3. In the latter case, [F2 + E5 ] is of the form [2L − Ei1 − Ei2 − Ei3 ], so [F 0 ] = [F3 + · · · + Fr ] + [G + 2L − Ei1 − Ei2 − Ei3 ], and we are done as above. Otherwise, [F] = [F1 + · · · + Fr + A1 + · · · + As + G 0 ], where each [Fj ] is in C4 , each [Aj ] is in C5 , and [G 0 ] ∈ hC0 ; C1 ; C3 i. Subtracting the class of a partial transform of a conic from each [Aj ] in [F] leaves [F1 + · · · + Fr ] + [G 00 ] with [G 00 ] in hC0 ; C1 ; C3 i. By the preceding argument and use of Corollary 2.3, S(F) = 0 and h1 (X; F) = 0.

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3. Six general points Throughout this section, let X be a blow-up of P2 at six general points p1 ; : : : ; p6 ; in particular, not all on a conic and no three on a line. We will determine s(H ) for any numerically eective divisor H on X . Lemma 3.1. The following classes: [L]; [2L − E1 − E2 − E3 ]; [3L − 2E1 − E2 − E3 − E4 − E5 ]; [4L − 2E1 − 2E2 − 2E3 − E4 − E5 − E6 ]; [H1 ] = [5L − 2E1 − 2E2 − 2E3 − 2E4 − 2E5 − 2E6 ]; [L − E1 ]; [2L − E1 − E2 − E3 − E4 ]; [H2 ] = [3L − 2E1 − E2 − E3 − E4 − E5 − E6 ]; and [3L − E1 − E2 − E3 − E4 − E5 − E6 ]; together with all classes obtained from these by permuting E1 ; : : : ; E6 ; generate the cone of numerically eective classes on X. Proof. Harbourne shows that every numerically eective class is in the cone generated by A = {[L]; [L − E1 ]; [2L − E1 − E2 ]; [3L − E1 − · · · − Ek ] | 3 ≤ k ≤ 6} under some Weyl transformation. (See Lemma III.1 in [13] for an indication of proof.) Applying all possible Weyl transformations to the elements of A gives a ÿnite list of classes; eliminating those which are sums of other classes in the list, we obtain the result. The classes [H1 ] and [H2 ] are named explicitly because we will refer to them often. Lemma 3.2. Let X be the blow-up of P2 at general points p1 ; : : : ; p6 . Then (a) There are only ÿnitely many exceptional classes on X; and these are the classes of the only reduced and irreducible curves of negative self-intersection on X. They are: [Ei ] for 1 ≤ i ≤ 6; [L − Ei − Ej ] for 1 ≤ i ¡ j ≤ 6; and [2L − Ei1 − · · · − Ei5 ] for 1 ≤ i1 ¡ · · · ¡ i5 ≤ 6: (b) Every eective class on X can be written as a sum of exceptional classes. (c) A divisor class [F] on X is numerically eective if and only if [F] · [C] ≥ 0 whenever [C] is an exceptional class on X. (d) If [F] is the class of a numerically eective divisor F on X; then F is eective; h1 (X; F) = 0; and the linear system |F| is base point (and hence ÿxed component) free.

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Proof. The lemma is well-known for six general points. The ÿrst statement of part (a), and part (c) are portions of [9, Theorem 8]. The second statement of (a) is shown in [14, Section 26]. Parts (b) and (d) are straightforward generalizations of [12, Lemma 3:1:1]. Lemma 3.3. Let [F] be a divisor class on X. Write [F] = [m0 L − m1 E1 − · · · − mn En ]. After reindexing if need be; we may assume m1 ≥ · · · ≥ mn . Then; since p1 ; : : : ; pn are general; [F] is numerically eective if and only if (i) mi ≥ 0 for all i; (ii) m0 ≥ m1 + m2 ; and (iii) 2m0 ≥ m1 + · · · + m5 . Proof. If [F] is numerically eective, [F] meets every eective class nonnegatively. In particular, it meets the eective classes [Ei ] (1 ≤ i ≤ 6); [L − E1 − E2 ]; and [2L − E1 − E2 − E3 − E4 − E5 ]. Conversely, the exceptional curves on X generate the cone of eective divisors, so if [F] meets every exceptional class nonnegatively, then [F] must be numerically eective. But every exceptional divisor on X is in a class of the form [Ei1 ], [L − Ei1 − Ei2 ] or [2L − Ei1 − Ei2 − Ei3 − Ei4 − Ei5 ], where the ik ’s within each class are distinct. Since [F] = [m0 L − m1 E1 − · · · − mn En ] with m1 ≥ · · · ≥ mn , conditions (i), (ii), and (iii) imply [F] meets every (−1)-curve nonnegatively. Theorem 3.4. Let X be the blow-up of P2 at six general points p1 ; : : : ; p6 . (Speciÿcally; we require that no three points are collinear and all six do not lie on a conic.) Let [H1 ] and [H2 ] be as above. Then (a) s(F) = 1 and r(F) = 0 for any divisor F in [H1 ]; (b) s(qF) = q and r(qF) = 0 if F is in [H2 ] or a corresponding class with permuted indices (for any q ≥ 0); and (c) s(F) = 0 if F is any other numerically eective class on X . Proof. Part (a) is shown in [13, Fact 6.2]. For part (b), let F ∈ [H2 ] and induct on q. If q = 0, the result is easy, so assume q ¿ 0. First we will show R(aF|H ) = 0 for all a ≥ 0, where H is a reduced irreducible eective divisor in [H2 ]. But R(aF|H ) is the kernel of 

H 0 (X; aF|H ) ⊗ H 0 (X; L) → H 0 (X; (aF + L)|H ): Since the ÿrst factor in the tensor product is 1-dimensional, it is enough to check that the map H 0 (X; L) → H 0 (X; L|H ) is injective. This last follows from the fact that H 0 (X; L − H2 ) = 0. Therefore  is injective, so R(aF|H ) = 0 for all a ≥ 0. Consider the exact sequences 0 → OX ((q − 1)F) → OX (qF) → OX (qF) ⊗ OH → 0 and 0 → OX ((q − 1)F + L) → OX (qF + L) → OX (qF + L) ⊗ OH → 0;

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for q ¿ 0. Taking global sections, and noting that h1 (X; (q−1)F) and h1 (X; (q−1)F +L) are 0 since (q − 1)F and (q − 1)F + L are both numerically eective, we see that Proposition 2.1 applies, yielding the exact sequence R((q − 1)F) → R(qF) → R(qF|H ): The argument above shows that the right-most term is 0, and the left-most term is 0 by induction. Therefore R(qF) = 0. Looking at the exact sequence 0 → H 0 (X; qF) ⊗ H 0 (X; L) → H 0 (X; qF + L) → S(qF) → 0 and using Riemann–Roch to compute dimensions, we ÿnd h0 (X; qF)=q+1; h0 (X; L)=3 and h0 (X; qF + L) = 4q + 3. Hence s(qF) = q, which completes the proof of (b). The proof of part (c) boils down to use of Theorem 2.6 and Corollary 2.3 in the following manner. Consider the following property for a divisor class [F]: (∗) [F] can be written as [F] = [H ] + [A1 + · · · + Ak ], where [H ] is a sum of divisors, each summand of which is numerically eective on a blow-up of P2 at ÿve or fewer points, each Ai is the proper transform of either a conic through ÿve of Pl p1 ; : : : ; p6 or a line through two of p1 ; : : : ; p6 , and [H ] + i=1 Ai is numerically eective for all 1 ≤ l ≤ k. If every element of a set
of divisor classes satisÿes (∗), then any class in the cone generated by
will also satisfy (∗). We point out that Theorem 2.6 together with Corollary 2.3 imply s(F) = 0 for any F which is in a class which satisÿes (∗), and for part (c) the classes in which we are interested form a submonoid of the cone of numerically eective classes. Thus we need only ÿnd a set of generators for this monoid, each element of which satisÿes (∗). We begin by letting
be the set consisting of the following classes, together with all classes obtained by permutations of E1 ; : : : ; E6 : [L]; [L − E1 ]; [2L − E1 − E2 − E3 ]; [2L − E1 − E2 − E3 − E4 ]; [3L − 2E1 − E2 − E3 − E4 − E5 ]; [3L − E1 − E2 − E3 − E4 − E5 − E6 ] = [L − E6 ] + [2L − E1 − E2 − E3 − E4 − E5 ]; [4L − 2E1 − 2E2 − 2E3 − E4 − E5 − E6 ] = [2L − E1 − E2 − E3 − E4 ] + [2L − E1 − E2 − E3 − E4 − E5 − E6 ]: Each element of
satisÿes (∗), so s(F) = 0 for every [F] in the cone generated by
, by the remarks above. Notice that the cone generated by
is the submonoid of the numerically eective cone containing all classes in which neither [H1 ] nor [H2 ] appear. We need to expand
to include multiples of [H1 ] and appropriate sums of numerically eective classes in which [H1 ] or [H2 ] is a summand. To accomplish the former, and prepare for the latter, we now replace
by the union of
and the following

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classes, together with the classes obtained by permuting indices: [2H1 ] = [10L − 4E1 − 4E2 − 4E3 − 4E4 − 4E5 − 4E6 ] = [2L − E3 − E4 − E5 − E6 ] + [2L − E1 − E2 − E4 − E5 − E6 ] + [2L − E1 − E2 − E3 − E5 − E6 ] + [2L − E1 − E2 − E3 − E4 − E6 ] + [2L − E1 − E2 − E3 − E4 − E5 ]; [3H1 ] = [15L − 6E1 − 6E2 − 6E3 − 6E4 − 6E5 − 6E6 ] = [L − E6 ] + [2L − E1 − E2 − E3 − E4 − E5 ] + [2L − E2 − E3 − E4 − E5 − E6 ] + [2L − E1 − E3 − E4 − E5 − E6 ] + [2L − E1 − E2 − E4 − E5 − E6 ] + [2L − E1 − E2 − E3 − E5 − E6 ] + [2L − E1 − E2 − E3 − E4 − E6 ] + [2L − E1 − E2 − E3 − E4 − E5 ]; [H2 + H20 ] = [6L − 3E1 − 3E2 − 2E3 − 2E4 − 2E5 − 2E6 ] = [2L − E1 − E2 − E5 − E6 ] + [2L − E1 − E2 − E3 − E4 − E6 ] + [2L − E1 − E2 − E3 − E4 − E5 ]; [H1 + H2 ] = [8L − 4E1 − 3E2 − 3E3 − 3E4 − 3E5 − 3E6 ] = [2L − E1 − E4 − E5 − E6 ] + [2L − E1 − E2 − E3 − E5 − E6 ] + [2L − E1 − E2 − E3 − E4 − E6 ] + [2L − E1 − E2 − E3 − E4 − E5 ]: The decompositions show that each new class satisÿes (∗). (Numerical eectivity at each stage is easily checked using Lemma 3.3.) Notice that the cone generated by
now contains all multiples of [H1 ], except [H1 ] itself. Next, check that the classes in the set [H1 ] +
= {[H1 ] + [G] | [G] ∈
} satisfy (∗). The following classes, together with those obtained by permutations of E1 ; : : : ; E6 make up [H1 ] +
: [6L − 2E1 − 2E2 − 2E3 − 2E4 − 2E5 − 2E6 ]; [6L − 3E1 − 2E2 − 2E3 − 2E4 − 2E5 − 2E6 ]; [7L − 3E1 − 3E2 − 3E3 − 2E4 − 2E5 − 2E6 ]; [7L − 3E1 − 3E2 − 3E3 − 3E4 − 2E5 − 2E6 ];

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[8L − 4E1 − 3E2 − 3E3 − 3E4 − 3E5 − 2E6 ]; [8L − 3E1 − 3E2 − 3E3 − 3E4 − 3E5 − 3E6 ]; [9L − 4E1 − 4E2 − 4E3 − 3E4 − 3E5 − 3E6 ]; [11L − 5E1 − 5E2 − 4E3 − 4E4 − 4E5 − 4E6 ]; [13L − 6E1 − 5E2 − 5E3 − 5E4 − 5E5 − 5E6 ]: Each of these classes may be decomposed (as was shown for [2H1 ]; [3H1 ], etc. above) to show that it satisÿes (∗). Thus we may replace
by the union of
and [H1 ] +
. At this point the cone generated by
includes all classes included in the statement of part (c) except for those of the form [F] = [qH20 ] + [G]; where [H20 ] = [3L − 2Ei1 − Ei2 − Ei3 − Ei4 − Ei5 − Ei6 ] and [G] is a nonzero class in the cone generated by
. We may assume [H20 ] = [H2 ]. We will use induction on q to show that each [F] of the indicated form satisÿes (∗). The case q = 0 is done, so assume q ¿ 0. Let [C]=[2L−E1 −E2 −E3 −E4 −E5 ] and let [Y ]=[L−E1 −E6 ]. Then [H2 ]=[C +Y ]. We claim that either [F − C] or [F − Y ] is numerically eective. Suppose not. Then [F − C] not numerically eective implies [F − C] must meet some eective class [E] negatively. Given that [F] = [qH2 + G] = [qC + qY + G], where [C] and [Y ] are exceptional classes and [G] is numerically eective, the only possibilities for [E] are [C] and [Y ]. Thus [F − C] · [C] ¡ 0 or ([F − C]) · [Y ] ¡ 0. Since [F] is numerically eective and [C] is a (−1)-curve, the former is impossible. Since [C] · [Y ] = 1, in order for the latter to hold, we must have [F] · [Y ] = 0. Similarly, [F − Y ] not numerically eective forces [F] · [C] = 0. But [F] = [qH2 ] + [G], where [G] is a nontrivial class in the cone generated by
. Clearly [H2 ] · [Y ] = 0 = [H2 ] · [C], so [G] · [Y ] = 0 = [G] · [C] as well. We know that [G] is in the cone generated by
, and it is easy to verify that the equalities [G] · [Y ] = 0 = [G] · [C] do not hold for any [G] in
, nor, consequently, for any nonzero [G] in the cone generated by this set. Thus either [F − C] or [F − Y ] must be numerically eective. Now [F] = [qH2 ] + [G] = [(q − 1)H2 ] + [G] + [Y ] + [C]. By the argument of the preceding paragraph, either (q − 1)[H2 ] + [G] + [Y ] or (q − 1)[H2 ] + [G] + [C] is numerically eective. Thus, by induction, one of the two satisÿes (∗), whence [F] does as well. Thus adding all elements of the form [F] = [qH2 ]0 + [G], where q ¿ 0 and [G] ∈
; [G] 6= 0, to
gives a generating set for the cone which consists of all numerically eective classes except [H1 ], nonnegative multiples of [H2 ], and permutations of the latter, as required. Corollary 3.5. If F is a numerically eective divisor on a general cubic surface X;  the multiplication map (X; F) ⊗ (X; L) → (X; F + L) has maximal rank.

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4. An example We begin by explaining the algorithm by which h0 ’s and Zariski decompositions can be determined, and then give an example of the computation of a minimal free resolution for a fat point ideal supported at six general points of P2 (see [7,8] for more details). The strategy for computing h0 (X; F) for a divisor F is to produce a divisor H , depending on F, such that h0 (X; H ) = h0 (X; F) and such that H is numerically eective if F is eective, and H · G ¡ 0 for some numerically eective divisor G if F is not eective. In the former case, h1 (X; H )=0 and [H ] is eective, so we compute h0 (X; H ) via Riemann–Roch, while in the latter case h0 (X; H )=0 since G is numerically eective. Here is the algorithm for determining H [8]. If F · Ei ¡ 0 for some 1 ≤ i ≤ n, then h0 (X; F) = h0 (X; F − (F · Ei )Ei ), so we may reduce to the case that F · Ei ≥ 0 for all 1 ≤ i ≤ n. Since L is numerically eective, if F · L ¡ 0, then F is not eective. Taking H = F, we are done (and h0 (X; F) = 0). If F · C ≥ 0 for every exceptional divisor C with C · L ¿ 0, then F is numerically eective and we take H = F. If F · C ¡ 0 for some exceptional divisor C with C · L ¿ 0, then h0 (X; F) = h0 (X; F − C), so we replace F by F − C and start over. Since there are only ÿnitely many exceptional classes to check (and it suces to check one exceptional divisor from each class) and (F −C)·L ¡ F ·L, the process terminates. To compute the number of elements of each degree in a minimal generating set for a fat point ideal, we will need the following fact [12]. Proposition 4.1. Let F be an eective divisor on a blow-up of P2 at six general points. If F = H + N; where H is the numerically eective divisor produced by the process above; then s(F) = s(H ) + h0 (X; F + L) − h0 (X; H + L). Let Ft = tL − m1 E1 − · · · − mn En . Let  be the smallest value of t for which h0 (X; Ft ) ¿ 0, and let be the smallest value of t for which H , as produced in the process described above, is exactly Ft . Example. Let p1 ; : : : ; p6 be general points of P2 and let Z = 12p1 + 9p2 + 6p3 + 6p4 + 6p5 + 3p6 . Setting Ft = tL − 12E1 − 9E2 − 6E3 − 6E4 − 6E5 − 3E6 , we have L IZ = t≥0 H 0 (X; Ft ). The divisor Ft ÿrst becomes eective in degree  = 18, and is numerically eective in degree = 21, so Ft is eective but not numerically eective only for d = 18; d = 19 and d = 20. We must write F18 ; F19 and F20 as sums of their numerically eective and ÿxed parts. The exceptional divisors L − E1 − E2 and 2L − E1 − E2 − E3 − E4 − E5 both meet F18 negatively, and are representatives of the only exceptional classes which meet the class of F18 negatively. We can subtract each divisor three times, giving the

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13

decomposition F18 = 9L − 6E1 − 3E2 − 3E3 − 3E4 − 3E5 − 3E6 | {z } H

+ 9L − 6E1 − 6E2 − 3E3 − 3E4 − 3E5 : | {z } N

The same exceptional classes meet the class of F19 negatively, and we can subtract a representative of the ÿrst class twice and of the second class once, giving the decomposition F19 = 15L − 9E1 − 6E2 − 5E3 − 5E4 − 5E5 − 3E6 | {z } H0

+ 4L − 3E1 − 3E2 − E3 − E4 − E5 : | {z } N0

Similarly, F20 = 19L − 11E1 − 8E2 − 6E3 − 6E4 − 6E5 − 3E6 + L − E1 − E2 : {z } | {z } | H 00

N 00

Using Proposition 4.1, Theorem 3.4, and Riemann–Roch to compute dimensions, we have s(Ft ) = 0

for t ¡ 17; 0

s(F17 ) = h (X; F18 ) = 4; s(F18 ) = s(H ) + h0 (X; F19 ) − h0 (X; H + L) = 3 + 19 − 15 = 7; s(F19 ) = s(H 0 ) + h0 (X; F20 ) − h0 (X; H 0 + L) = 0 + 39 − 36 = 3; s(F20 ) = s(H 00 ) + h0 (X; F21 ) − h0 (X; H 00 + L) = 0 + 61 − 60 = 1; s(Ft ) = 0

for t ¿ 20:

Thus a minimal generating set for IZ requires 15 elements (four of degree 18, seven of degree 19, three of degree 20, and one of degree 21), and the ÿrst module F0 in a minimal free resolution 0 → F1 → F0 → IZ → 0 for IZ is F0 = R[ − 18]4 ⊕ R[ − 19]7 ⊕ R[ − 20]3 ⊕ R[ − 21], where R = k[P2 ]. The Hilbert functions of F0 ; IZ and F1 are given by   X t+2−n s(Fn−1 ) hF0 (t) = 2 n         t − 16 t − 17 t − 18 t − 19 +7 +3 + ; =4 2 2 2 2 hIZ (t) = h0 (X; Ft ); hF1 (t) = hF0 (t) − hIZ (t): We give a table of values for all three.

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S. Fitchett / Journal of Pure and Applied Algebra 156 (2001) 1–14

t

17

18

19

20

21

22

23

24

25

26

···

hF0 (t) hIZ (t) hF1 (t)

0 0 0

4 4 0

19 19 0

48 39 9

92 61 31

151 84 67

225 108 117

314 133 181

418 159 259

537 186 351

··· ··· ···

Working backwards to ÿnd F1 = R[ − 20]9 ⊕ R[ − 21]4 ⊕ R[ − 22], we have a minimal free resolution for IZ : 0 → R[ − 20]9 ⊕ [ − 21]4 ⊕ R[ − 22] → R[ − 18]4 ⊕ R[ − 19]7 ⊕ R[ − 20]3 ⊕ R[ − 21] → IZ → 0: Acknowledgements The results in this paper form part of my doctoral dissertation. I wish to express my sincere appreciation to my advisor, Brian Harbourne, for many illuminating discussions and helpful suggestions. References [1] M.V. Catalisano, “Fat” points on a conic, Comm. Algebra 19 (8) (1991) 2153–2168. [2] L. Ein, R. Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993) 51– 67. [3] S. Fitchett, Generators of fat point ideals on the projective plane, Doctoral Dissertation, University of Nebraska, Lincoln, 1997. [4] F.J. Gallego, B.P. Purnaprajna, Higher syzygies of elliptic ruled surfaces, J. Algebra 186 (1996) 626–659. [5] M. Green, Koszul cohomology and the geometry of projective varieties, J. Dierential Geom. 19 (1984) 125–171. [6] B. Harbourne, Blowings-up of P2 and their blowings-down, Duke Math. J. 52 (1985) 129–148. [7] B. Harbourne, Complete linear systems on rational surfaces, Trans. A.M.S. 289 (1985) 213–226. [8] B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Can. Math. Soc. Conf. Proc. 6 (1986) 95–111. [9] B. Harbourne, Rational surfaces with K 2 ¿ 0, Proc. A.M.S. 124 (1996) 727–733. [10] B. Harbourne, Anticanonical rational surfaces, Trans. A.M.S. 349 (1997) 1191–1208. [11] B. Harbourne, An algorithm for fat points on P2 , preprint, 1997. [12] B. Harbourne, Free resolutions of fat point ideals on P2 , J. Pure Appl. Algebra 125 (1998) 213–234. [13] B. Harbourne, The ideal generation problem for fat points, J. Pure Appl. Algebra 145 (2000) 165–182. [14] Y.I. Manin, Cubic Forms, North-Holland Mathematical Library, vol. 4, North-Holland, Amsterdam, 1986. [15] D. Mumford, Varieties deÿned by quadratic equations, Questions on algebraic varieties, Corso C.I.M.E. Cremoneses, Rome, 1969, pp. 30 –100.